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Two Bayesian models with different sampling densities are said to be marginally equivalent if the joint distribution of observables and the parameter of interest is the same for both models. We discuss marginal equivalence in the general framework of group invariance. We introduce a class of sampling models and establish marginal equivalence when the prior for the nuisance parameter is relatively invariant. We also obtain some robustness properties of invariant statistics under our sampling models. Besides the prototypical example of $v$-spherical distributions, we apply our general results to two examples---analysis of affine shapes and principal component analysis.
An important problem in large scale inference is the identification of variables that have large correlations or partial correlations. Recent work has yielded breakthroughs in the ultra-high dimensional setting when the sample size $n$ is fixed and t
We consider the asymptotic behaviour of the marginal maximum likelihood empirical Bayes posterior distribution in general setting. First we characterize the set where the maximum marginal likelihood estimator is located with high probability. Then we
We introduce a general method, named the h-function method, to unify the constructions of level-alpha exact test and 1-alpha exact confidence interval. Using this method, any confidence interval is improved as follows: i) an approximate interval, inc
The existence of the maximum likelihood estimate in hierarchical loglinear models is crucial to the reliability of inference for this model. Determining whether the estimate exists is equivalent to finding whether the sufficient statistics vector $t$
Multivariate distributions are explored using the joint distributions of marginal sample quantiles. Limit theory for the mean of a function of order statistics is presented. The results include a multivariate central limit theorem and a strong law of